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In this paper, we consider an unmanned aerial vehicle (UAV) aided mobile relaying system under a buffer constraint. We propose a new relaying protocol employing mixed free-space optical/radio frequency (FSO/RF) communication, i.e., the source-relay and relay-destination links utilize FSO and RF links, respectively, under the buffer constraint at the UAV relay node. Taking the conditions of an imbalance in transmission rate between RF and FSO links into consideration, we study the trajectory optimization problem of buffer-constrained UAV relay node in order to maximize the end-to-end data throughput. Especially, we classify two relaying transmission schemes according to the delay requirements, i.e., i) delay-limited transmission and ii) delay-tolerant transmission. We solve the locally optimal trajectory problem of the UAV to maximize the throughput of ground user terminal. As a result, we propose an iterative algorithm that efficiently finds a local optimum solution for the throughput maximization problems. Through this algorithm, we present the resulting trajectories over the the atmospheric condition, the buffer size, and the delay requirement. Also, we show the optimum buffer size and the throughput-delay tradeoff for a given system. Our numerical results validate that the proposed buffer-aided mobile relaying scheme achieves 65.55% throughput gains compared to conventional static relaying scheme.